Find the maximum value of $\det(M)+ \det(N)$?

Question

$M,N $ are two $4\times 4 $ matrices satisfying, $$ MN= \begin{bmatrix} 2 & 0 & 0 & 2 \\ 0 & 2 & 2 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{bmatrix} $$ I have no idea how to approach this problem, help will be appreciated.

Find the maximum value of $\det(M)+ \det(N)$

Answer 1

Hint:$$\bigl(\forall x\in(0,\infty)\bigr):(x\operatorname{Id}).\begin{bmatrix}\frac{2}{x} & 0 & 0 & \frac{2}{x} \\ 0 & \frac{2}{x} & \frac{2}{x} & 0 \\ 0 & 0 & \frac{2}{x} & 0 \\ 0 & 0 & 0 & \frac{2}{x}\end{bmatrix}=\begin{bmatrix}2 & 0 & 0 & 2 \\ 0 & 2 & 2 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2\end{bmatrix}.$$

Answer 2

Let $A = MN$ then we can calculate the minimum value, if $\det(M)>0$: $$\det (M)+\det (N) = \det(M)+{\det(A) \over \det(M)}\geq 2\sqrt{\det(A)} = 8$$

but there is no maximum value for $$\det(M)+{16 \over \det(M)}$$

Just draw a function $f(x) = x+{16\over x}$ where $x=\det(M)\in\mathbb{R}\setminus\{0\}$